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The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_{un})$ over the field with one element $\mathbb{F}_{un}$ is, simply by taking $q=1$, $[n]_1=n$. Similarly, whenever the number of $\mathbb{F}_q$ points on a variety is a polynomial in $q$, we can take the $q=1$ limit just as well.

On the flip side of the coin, the $q$-integers appear in many $q$-analogs, including representation theory of quantum groups. In this context, $q$ is often a root of unity $q=e^{\frac{2\pi i}{k}}$ or more generally a complex number, and $q\rightarrow 1$ limit is understood as a "classical limit".

The connection is probably not superficial, because there are other instances where $q$ is interpreted in two different ways. For instance, according to this slide, it's a theorem of Katz that for a smooth quasi-projective variety $X$ defined over $\mathbb{Z}$, if the number of $\mathbb{F}_q$ points is a polynomial in $q$, then it is the E-polynomial of $X$, which is a specialization of the weight polynomial $WH(X;q,t)$. On the other hand, Chuang-Diaconescu-Pan related the weight polynomial to the refined Gopakumar-Vafa expansion.

So, my question is, is there any simple explanation of the apperance of $q$ in two different guises, a power of a prime and a root of unity?

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    $\begingroup$ Somehow, prime powers and roots of unity feel "dual" to each other. Perhaps this can be made precise through the fact that as locally compact abelian groups, $\mathbb{Z}$ and $\mathbb{S}^1$ are dual to each other, but don't ask me how to relate this fact to the examples that you mention... $\endgroup$ – Tom De Medts Jan 22 at 9:27
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This is very not rigorous, but it's a way of thinking about this topic which I find personally helpful. Several $\mathbb F_1$ papers contains remarks about the idea going back to Weil and Iwasawa that adjoining roots of unity is analogous to base field extensions. This means that $\mathbb F_1,\mathbb F_{1^2},\dots$ shouldnt be thought of as the same "$F_{un}$", rather you should think of $\mathbb F_{1^n}$ as $\mathbb F_1$ together with $n$-th roots of unity. Notice that the number of $\mathbb F_1$ points will be the same as the number of $\mathbb F_{1^n}$ points since plugging in $q=1$ or $1^n$ gives the same thing, however things get more interesting when considering the relation between the fields rather than individually.

Just like $\mathbb F_q$ points are the Frobenius fixed points of $\mathbb F_{q^n}$, in combinatorics land $\mathbb F_1$ points should be the points of a $\mathbb F_{1^n}$ scheme fixed by a cyclic group of order $n$. Just like how you expect to count $\mathbb F_1$ points by plugging in $q=1$ formally when the point count is uniform over all finite fields, counting $\mathbb F_1$ points in a variety over $\mathbb F_{1^n}$, should have something to do with plugging in formally $q^n=1$, or in other words some primitive $n$-th root of unity. This perspective makes cyclic sieving natural in many combinatorial contexts.

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One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $k$ of characteristic $p$ not dividing q, so $q$ can be simultaneously a root of unity and a prime power.

Let's say we are in type $A$, then the Iwahori-Hecke algebra $\mathcal{H}_q(n)$ and the $q$-Schur algebra $S(q,n)$ over $k$ arise as endomorphism algebras of certain unipotent representations of $GL_n(\mathbb{F}_q)$ and control many aspects of the unipotent representations, in fact there is an equivalence of categories between $S(q,n)$-modules and unipotent representations (for $\mathcal{H}_q(n)$-modules there are biadjoint functors, but not equivalences in general).

For example we can see that if the order of $q$ (mod p) is bigger than $n$ then the order of $GL_n(\mathbb{F}_q)$ is prime to $p$ and hence every representation is completely reducible. This corresponds to the fact in characteristic zero that the algebras $\mathcal{H}_q(n)$ and $S(q,n)$ fail to be semisimple exactly when $q$ is a root of unity of order less than $n.$

One can relate the cases of positive characteristic and characteristic zero using model theory. In particular if you let the characteristic $p$ tend to infinity while always choosing $q$ to be a primitive $d$th root of unity then the representation theory of $S(q,n)$ (or $\mathcal{H}_q(n)$) in characteristic $p$ "converges" (in a model theoretic sense) to that of $S(\zeta,n)$ in characteristic zero (where $\zeta$ is a primitive $d$th root of unity).

Hence you can prove things about Schur algebras and Iwahori-Hecke algebras at roots of unity in characteristic zero (and therefore about quantum groups) by looking at unipotent representations of $GL_n(\mathbb{F}_q)$ in non-describing characteristic. For example it's not too hard to turn the above paragraph into an actual proof of for which values of $q$ these algebras are semisimple in characteristic zero.

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